Just a theory ...
Please comment.

Afro-Asiatic spread already in mesolithic times along with E1b1b1, into the Levant, along the North African shores and into northeastern Africa.
The Natufians and the Levant Neolithic were E1b1b1 supplemented with some H2.
The Anatolian farmers were mainly G2a.
J is only marginal in the neolithic, and probably also in chalcolithic.
The bronze metallurgy spread from eastern Anatolia or Mesopotamia into the Levant and the Aegean, and J became the major haplogroup.
Semitic was an Afro-Asiatic bronze age language in the Levant, and it was also the language of the Akkadians who arrived in southern Mesopotamia prior to written records. They were newcomers from the north though, pushing the Sumerians south.
In the Aegean and bronze age Anatolia the main haplogroup was the newcomer J, but autosomal the population was still neolithic.
J was probably a male elite with local wives.
And the male J didn't raise his children. He left it to the mothers, and they taught them the local languages.